If and , then show that A2 = B2 = C2 = I2.

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Special Type of Square Matrices

Question

If $A = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}], B = [\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}]$ and $C = [\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}]$ , then show that A² = B² = C² = I₂.

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$A = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] B = [\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}] C = [\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}]$

A^{2} = [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}]

= [\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}] = I_{2}

__(1)

B^{2} = [\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}] [\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}]

[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = I_{2}

__(2)

C^{2} = [\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}] [\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}]

= [\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}] = I_{2}

__(3)

From (1) (2) and (3) we get

A² = B² = C² = I₂.